This is the end of the preview. Sign up
to
access the rest of the document.

Unformatted text preview: Harmonic Waves A wave generated by the simple harmonic vibrations of the particles of a medium is called a harmonic wave . What is the equation for the displacement of a simple harmonic motion? At t = 0 if the particle is at the equilibrium position, the displacement y is: y = A sin ( t) ince a harmonic wave is formed by the varying Since a harmonic wave is formed by the varying displacements of the vibrating particles, the equation for a harmonic wave is: y = A sin ( t) 2 2 T f = = 2 sin y A t T = 1 A is the amplitude of the wave and f is the frequency. The displacement y can also be written as a function of x 2 sin y A t T = v f = [ ] sin 2 A ft = sin 2 v y A t = n 2 si vt A = [ ] sin y A x k = is called 2 wave number the k = 2 To describe a wave advancing to the right, we replace x by x vt where v is the wave speed. [ ] sin y x A k = The equation for a wave advancing to the right is: y = A sin[k(x vt)] 2 ) ( sin x vt y A =- 2 2 sin A t v x - = = A sin ( k x 2 f t) y = A sin (kx t) Similarly a wave advancing to the left can be written as: y = A sin[k(x + vt)] y = A sin (kx + t) http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Fla sh/ClassMechanics/TravelWaves/TravelWaves.html 3 The wave y(x,t) = A cos k(x + 34 t) represent a traveling wave where x is in meters and t in second. What are the direction and the speed of the wave? Sine function or cosine function only determines the initial state of the wave and not the direction. Example 1 The wave y(x,t) = A sin k(x + v t) is a wave traveling to the left. The wave y(x,t) = A cos k(x + 34 t) is a wave traveling to the left with a speed equal to 34 ms-1 . 4 At time t = 0, the shape of a wave pulse on a string is given by the function: where x is in meters. Plot y(x,0) versus x and give the general wave function y(x,t) at any time t if (i) the pulse is moving in the positive x-direction with a speed of 10...
View Full
Document